Drawing Graph of the Function y=f(kx)

Task 1. Draw graph of the function `y=f(kx)`, where `k>0,k!=1`, knowing graph of the function `y=f(x)`.

Let `y_0=f(x_0)`. Now answer the following question: what value of argument `x` should we take, so function `y=f(kx)` will take value `y_0`? Clearly this value should satisfy the following condition: `kx=x_0` or `x=(x_0)/k`. Therefore, point `(x_0;y_0)` that lies on graph of the given function `y=f(x)` is trasformed into point `((x_0)/k;y_0)` that lies on the graph of the function `y=f(kx)`. This transformation is called compressing of graph `y=f(x)` with coeffcient k to y-axis (if 0<k<1 then, in fact, we stretch from y-axis with coeffcient `1/k`).

On figure to the right you can see graphs of the following functions: `y=arccos(x)` and `y=arccos(2x)`; `y=sqrt(x)` and `y=sqrt(x/3)`. In first case, we compress graph to y-axis because k=2>1; in second case we stretch graph because `0<k=1/3<1`.

Let `y_0=f(x_0)`. Function `y=f(-x)` will take value `y_0` if argument `x` satisfies following condition: `x_0=-x`, i.e. `x=-x_0`. Point of the graph `(x_0;y_0)` of graph of the function `y=f(x)` is transformed into point `(-x_0;y_0)` of graph of the function `y=f(-x)`. This means that we can obtain graph of the function `y=f(-x)` by reflecting graph of the function `y=f(x)`about y-axis.

On figure to the left you can see graphs of functions `y=log_3(x)` and `y=log_3(-x)`.

Task 3. Draw graph of the function `y=f(kx)`, where `k<0,k!=-1`, knowing graph of the function `y=f(x)`.

We have that `f(kx)=f(-|k|x)`. Therefore, to obtain graph of the function `y=f(kx)` we first compress graph of the function `y=f(x)` to y-axis with coeffcient `|k|` and the reflect result about y-axis.

On the figure to the right you can see graphs of function `y=x^(3/2)` and `y=(-2x)^(3/2)`.